rizer.electric_circuit.nrp_circuit#

Classes#

NRPCircuit

Nanosecond Repetitive Pulsed (NRP) discharge circuit.

Module Contents#

class rizer.electric_circuit.nrp_circuit.NRPCircuit(generator: rizer.electric_circuit.generator.PurelyResistiveBaseGenerator, cable: rizer.electric_circuit.cable.IdealCable)#

Nanosecond Repetitive Pulsed (NRP) discharge circuit.

The circuit models a pulsed voltage generator connected to a plasma load through an ideal transmission line. It is used for sub-microsecond pulsed plasma applications where wave propagation and multiple reflections along the cable matter.

Topology:

[Generator] ----[Ideal cable]----[Plasma load :math:`R_p(t)`]
   :math:`R_g`         :math:`Z_c, L, c`
   :math:`V_g(t)`

The generator has a purely resistive internal impedance \(R_g\) and a time-dependent open-circuit voltage \(V_g(t)\). The cable is lossless, with characteristic impedance \(Z_c\), length \(L\), and wave speed \(c\). The plasma is represented as a time-varying resistance \(R_p\) at the far end of the line.

At the generator–cable junction:

\[\alpha_g = \frac{Z_c}{Z_c + R_g}, \qquad \Gamma_g = \frac{R_g - Z_c}{R_g + Z_c}\]

where \(\alpha_g\) is the voltage attenuation coefficient and \(\Gamma_g\) is the reflection coefficient. The plasma-end reflection coefficient \(\Gamma_p\) is computed by Γ_p().

The plasma terminal voltage is computed with the SPICE-style traveling-wave recursion (Branin’s method of characteristics for an ideal line, [Branin1967]): the incident wave at the plasma end is

\[a(t) = \alpha_g V_g(t) + \Gamma_g \, b(t - \tau), \qquad b(t) = \Gamma_p(t) \, a(t), \qquad V_p(t) = (1 + \Gamma_p)\, a(t)\]

with \(\tau = 2L/c\). Only one round trip of the reflected wave is buffered; unrolling the recursion reproduces the bounce-diagram Neumann series to infinite order, so all reflections are included exactly and nb_reflections is not needed (it is honored only as the 0 special case, direct wave only). O(1) work per evaluation, O(round-trip) memory. The recursion is a contraction (\(|\Gamma_g \Gamma_p| < 1\) for any passive load), so interpolation errors decay instead of accumulating.

Parameters:

References

[Branin1967]

F. H. Branin, Transient analysis of lossless transmission lines, Proc. IEEE 55 (1967) 2012-2013.

See also

PurelyResistiveBaseGenerator, IdealCable

generator#
cable#
R_g#
Z_c#
alpha_g#
Γ_g#
round_trip_time#

Cable round-trip time \(\tau = 2L/c\) [s].

compute_plasma_voltage(t: float, R_p: float, nb_reflections: int) → float#

Plasma terminal voltage including cable reflections.

The incident wave from the generator is attenuated by \(\alpha_g\) and may reflect back and forth between the generator (\(\Gamma_g\)) and the plasma (\(\Gamma_p\)) ends. Each round trip introduces a delay \(\tau = 2L/c\).

For nb_reflections == 0, only the first forward wave is retained:

\[V_p(t) = \alpha_g \, V_g(t)\]

Otherwise, the traveling-wave recursion (see NRPCircuit) includes all reflections exactly regardless of the value of nb_reflections — it is honored only as the 0 special case above. The result is multiplied by the transmission coefficient \(2 R_p / (R_p + Z_c)\).

Each call appends the reflected wave at t to the internal one-round-trip buffer, so it can be interpolated when evaluating the recursion at a later time.

Parameters:

  • t (float) – Time, in seconds.

  • R_p (float) – Plasma resistance at time t, in Ohm.

  • nb_reflections (int) – Number of round-trip reflections to include. Use 0 for the direct (non-reflected) contribution only.

Returns:

Plasma terminal voltage, in Volts.

Return type:

float

Raises:

  • TypeError – If R_p is not numeric or nb_reflections is not an integer.

  • ValueError – If R_p or nb_reflections is negative.

Γ_p(R_p: float) → float#

Reflection coefficient at the plasma end.

Parameters:

R_p (float) – Plasma resistance in Ohm.

Returns:

Reflection coefficient at the plasma end, dimensionless.

Return type:

float

Notes

The reflection coefficient at the plasma end is given by:

\[\Gamma_p = \frac{R_p - Z_c}{R_p + Z_c}\]

where:

  • \(R_p\) is the plasma resistance in Ohm,

  • \(Z_c\) is the cable characteristic impedance in Ohm.